Download E 243 Spring 2015 Lecture 1

Announcements
Organization of the course – 4 quizzes, 5* homeworks, and Final
Important Dates on Syllabus sheet
Textbook – Probability and Statistics for Engineers and Scientists, Anthony Hayter, 4th
edition, Lecture Notes by Dr. Sarath Jagupilla
HW # 1 problems – 1.2.11, 1.3.14, 1.4.10, 1.5.18, 1.6.10, 1.7.8, 1.7.18
HW#1 Due on February 2, 2015
1. Probability
Probability is a measure of confidence that an event will occur. The probability of an
event A is denoted as P(A). The complement of an event A is the collection of all
outcomes that are not contained in event A. It is denoted by Ac.
𝑃(𝐴𝑐 ) = 1 − 𝑃(𝐴)
An event is any collection of outcomes of an experiment. An experiment is any process
that generates more than one outcome. The collection of all possible outcomes of an
experiment is the sample space for that experiment.
Sets and Axioms of Probability
Probability, as are many other fields in mathematics, is based on set theory. Given the
context of an experiment, a set is any collection of its outcomes. Venn diagrams,
conceived by John Venn in 1880, are great tools to learn simple relationships between
sets. A Venn diagram usually consists of a rectangle which represents the sample space,
and any closed shape within the rectangle which represents an event.
Figure 1-1 – Venn Diagram
From the Venn diagram it is clear that the event is always a subset of the sample space.
The two extremes are that it can either occupy the entire sample space or the event has
no possible occurrences in the sample space. These observations are captured in the
following axioms of probability.
1) The probability of the sample space, S, is one i.e., P(S) = 1
2) The probability of an event that has no possible occurrences is zero i.e., 𝑃(∅) = 0
3) The probability of any event A always lies between zero and one i.e., 0 ≤ 𝑃(𝐴) ≤
1
Unions and Intersections
Some basic relationships could be developed between two sets using Venn diagrams.
Consider two sets representing the events A and B as shown,
Figure 1-2 – Unions and Intersections
The intersection of the two events A and B is the collection of all outcomes that occur in
both the events simultaneously. The probability of the intersection is denoted by 𝑃(𝐴 ∩
𝐵).
The union of the two events A and B is the collection of all outcomes that occur in at
least one of the two events. The probability of the union is denoted by 𝑃(𝐴 ∪ 𝐵). The
relationship between the union and intersection of two events is given by,
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃𝐴 ∩ 𝐵)
We subtract the intersection once because we double counted it when we added the
probabilities of A and B.
Mutually Exclusive and Exhaustive Events
Two events are said to be mutually exclusive if they don’t have any common elements.
This implies that their intersection, 𝐴 ∩ 𝐵, is empty and has no events. So, two events
are mutually exclusive if,
𝑃(𝐴 ∩ 𝐵) = 0
Figure 1-3 – Mutually Exclusive Events
Further, if two events are mutually exclusive their union is just the sum of individual
probabilities, because the probability of the intersection is zero.
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
This could be extended to any number of events,
𝑃(𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 ) = 𝑃(𝐴1 ) + 𝑃(𝐴2 ) + ⋯ + 𝑃(𝐴𝑛 )
Two events are said to be exhaustive if, together, they occupy the entire sample space.
So, two events are exhaustive if,
𝑃(𝐴 ∪ 𝐵) = 1
Figure 1-4 – Exhaustive Events
This can be extended to any number of events. So events A1, A2, … , An are exhaustive if,
𝑃(𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 ) = 1
Short Review – Probability, Experiment, Event, Complement of an Event, Axioms of
Probability, Unions and Intersections (relation between them), Mutually Exclusive
Events, Exhaustive Events.
Conditional Probability, Independence, Baye’s Rule, and Theorem of Total
Probability
Sometimes, we come to know that an event has already happened and we want to
incorporate this fact into our probability calculations. Given that event B has already
happened, the probability of A is termed conditional probability of A given B and it is
calculated as,
𝑃(𝐴/𝐵) =
𝑃(𝐴 ∩ 𝐵)
𝑃(𝐵)
Figure 1-5 – Conditional Probability
The formula could be understood by examining the Venn diagram.
Similarly, the conditional probability of event B given that A has occurred is given by,
𝑃(𝐵/𝐴) =
𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴)
The two equations can be combined and rearranged to result in the Baye’s rule,
𝑃(𝐴/𝐵) =
𝑃(𝐵/𝐴). 𝑃(𝐴)
𝑃(𝐵)
Two events are said to be independent of each other if the occurrence of either event
does not affect the occurrence of the other event. So, whether event B happens or not, it
would not affect the probability of event A. Therefore if A is independent of B then,
𝑃(𝐴/𝐵) = 𝑃(𝐴)
Using the conditional probability rule in the definition of independence, a condition for
independence is given by,
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵)